Knowledge-based expedited parameter tuning of microwave passives by means of design requirement management and variable-resolution EM simulations

The importance of numerical optimization techniques has been continually growing in the design of microwave components over the recent years. Although reasonable initial designs can be obtained using circuit theory tools, precise parameter tuning is still necessary to account for effects such as electromagnetic (EM) cross coupling or radiation losses. EM-driven design closure is most often realized using gradient-based procedures, which are generally reliable as long as the initial design is sufficiently close to the optimum one. Otherwise, the search process may end up in a local optimum that is of insufficient quality. Furthermore, simulation-based optimization incurs considerable computational expenses, which are often impractically high. This paper proposes a novel parameter tuning procedure, combining a recently reported design specification management scheme, and variable-resolution EM models. The former allows for iteration-based automated modification of the design goals to make them accessible in every step of the search process, thereby improving its immunity to poor starting points. The knowledge-based procedure for the adjustment of the simulation model fidelity is based on the convergence status of the algorithm and discrepancy between the current and the original performance specifications. Due to using lower-resolution EM simulations in early phase of the optimization run, considerable CPU savings can be achieved, which are up to 60 percent over the gradient-based search employing design specifications management and numerical derivatives. Meanwhile, as demonstrated using three microstrip circuits, the computational speedup is obtained without design quality degradation.

Geometries of microwave passive components become continuously more involved to fulfil the performance requirements of various application areas (wireless communications including 5G and 6G 1,2 , internet of things 3 , wireless sensing 4 , microwave imaging 5 , wearable devices 6 , autonomous vehicles 7 , etc.). In particular, many of these applications require specific functionalities such as multi-band operation 8 , harmonic suppression 9 , customized phase characteristics 10 , reconfigurability 11 , often combined with limitations on the physical size of the devices [12][13][14][15] . Due to their inherent complexity, microwave structures designed to meet these and other requirements typically feature considerably increased numbers of design variables than conventional circuits, whereas their design process has to account for several objectives and constraints imposed on their electrical characteristics. Adequate parameter tuning of such circuits requires rigorous numerical optimization [16][17][18][19][20] . At the same time, the adjustment process has to be carried out at the level of full-wave electromagnetic (EM) simulations to account for effects such as EM cross-couplings, or dielectric and radiation losses. These cannot be properly quantified using analytical or equivalent network models, yet are important for the operation of modern circuits implemented using techniques such as transmission line folding 21 , defected ground structures 22 , the employment with U being a merit function (or objective function). For additional clarification, consider an equal-split coupler, which is to run at the operating frequency f 0 ; the device is supposed to minimize both input matching and port isolation at f 0 . Given the above, the characteristics of interest are S-parameters S j1 , j = 1, …, 4. The function U can take the form of Note that the objectives are categorized: minimization of |S 11 | and |S 41 | is the primary goal; the equal power split condition is an equality constraint, ensured by the penalty term proportional to the penalty coefficient β. It should be emphasized that (2) is only an illustrative example, whereas the overall concept outlined here is generic and applicable to other EM-driven tasks. Figure 1 shows the example of a branch-line coupler, which is intended to work at 1.8 GHz. Local search starting from the design indicated using black lines will succeed, whereas optimization from the design shown using the grey lines will fail because of a significant discrepancy between the target and existent operating frequencies.
The automated adaptive specification management scheme 70 addresses the above issue by relocating the targets throughout the optimization run so that they are reachable at each stage of the process. The amount of relocation depends on the detected discerned discrepancy between the existent and desired operating frequencies. A graphical illustration of the specification management procedure has been provided in Fig. 2.
Automated adaptation of design requirements: prerequisites and implementation. The design specification management scheme operates based on the following prerequisites: (i) the necessary relocation of the target frequencies should be identified using the actual operating conditions at the current design, and (ii) the relocated goals should be reachable through local search.
In the following, J(x) will represent the sensitivity matrix of the system outputs S(x). Assuming that the optimization procedure is iterative and yields approximate designs x (i) , i = 0, 1, …, to the optimum solution x * of (1) (x (0) denotes the starting point), we utilize the first-order linear expansion model L S www.nature.com/scientificreports/ Also, consider an auxiliary sub-problem where the search radius D is typically set to D = 1, whereas x tmp denotes the temporary design. The automated alteration of design specs is based on the factors described in Table 1 70 .These guide the decision-making process and are used to define the conditions gathered in Table 2, satisfaction of which decides upon modification of the design goals with respect to their original values aggregated in the target vector F. In particular, satisfaction of any of the conditions is considered an indication that the performance requirements are unlikely to be attained and should be relaxed accordingly.

Improvement factor
S (x (i) ), F) Determines potential for design improvement starting from x (i) Distance between the actual and target operating frequencies Used as a safeguard to ensure that the updated specifications are sufficiently close to the current operating frequencies www.nature.com/scientificreports/ The thresholds F r.min (for improvement factor) and D c.max (for distance between the actual and target operating frequencies) of Table 2 are generally problem dependent, and should be set up having in mind the typical (or expected) operating bandwidths of the circuit at hand. A simple procedure for adjusting these values has been described in Koziel et al. 70 .
Having defined the decision factors and adjustment conditions, we can now summarize the design specification management procedure. The target operating frequencies for the (i + 1)th iteration of the optimization algorithm will be denoted as F current (a) = [f current.1 (a) … f current.N (a)] T , where 0 ≤ a ≤ 1. The individual frequencies f current.k are obtained as where f c.k are the existent operating frequencies at x (i) (we also have the vector of current operating frequen- . The value of the factor a is determined as the maximal value a ≤ 1 so that F r ≥ F r.min and D c ≤ D c.max at the design x tmp obtained through minimizing (cf. (4)) In practice, a is found using an auxiliary numerical optimization process, in which it is gradually reduced (starting from a = 1) so that F r ≥ F r.min and D c ≤ D c.max for x tmp produced by (6). Satisfaction of both conditions means that the current targets are reachable from x (i) . In the course of the optimization run, the adjusted specs will ultimately converge to the assumed values (which is equivalent to satisfying both F r ≥ F r.min and D c ≤ D c.max for a = 1), assuming that that initial specs are attainable. Otherwise, the algorithm will terminate when getting as close to the targets as achievable.
The described decision-making procedure is executed before each iteration of the search process. Consequently, the design targets are continuously adjusted to account for the current discrepancies between the actual and desired operating parameters. An important observation is that the modification process incurs no extra computational costs (in terms of additional EM evaluations), because it is based on the sensitivity data already evaluated during routine working of the optimization procedure.

Variable-fidelity models for optimization cost reduction
In this work, the primary tool incorporated to enhance computational efficacy of the optimization process is the incorporation of variable-fidelity EM simulation models. This section explicates the introduced knowledge-based model fidelity management procedure, which is based on two factors: (1) the detected discrepancy between the current and original design targets, and (2) the convergence indicators of the algorithm.

Variable-fidelity EM simulations.
Computational models of microwave devices can be implemented using full-wave EM analysis 71 , or circuit theory tools (equivalent networks 72 , analytical descriptions 73 ). In this work, it is assumed that the primary (high-fidelity) representation of the circuit of interest is in the form of (highfidelity) EM simulation, whereas lower-fidelity models are obtained through EM analysis executed at lower discretization density of the structure. This is a versatile and easy to control way, which also ensures a sufficiently good correlation between the models of different resolutions. Other simplification factors (e.g., neglecting losses, reducing computational domain 74 ) will not be considered here. Utilization of multi-fidelity models can be beneficial for computational efficiency of the CAD procedures, e.g., space mapping 75 , response correction 45,46 , cokriging 76 . Typically, two levels of fidelity are employed (coarse/low-fidelity, fine/high-fidelity) 77 ), which raises some practical issues related to appropriate model selection and setup 78 .
Model fidelity can be modified using the parameters controlling the meshing algorithms, e.g., lines-perwavelength (LPW) of CST Microwave Studio. Figure 3 provides an example of a dual-band power divider and the relationship between LPW and the average simulation times of the structure. The minimum model fidelity (here, denoted as L min ) should be selected to ensure that the corresponding circuit responses are still representative, i.e., not excessively distorted with respect to the maximum fidelity (here, denoted as L max ). The latter, corresponding to the high-fidelity model, should render the circuit response of the accuracy considered sufficient for practical purposes. Observe that in our work, the model accuracy is understood as the accuracy of the EM simulation model implemented is CST Microwave Studio, and it depends on the mesh density. In general, the higher the density, the better the model accuracy.
The computational model fidelity L is selected from the admissible range L min ≤ L ≤ L max . At the initial phase of the optimization procedure, L is set to L min so as to accelerate the optimization process. Towards the end of the run, L gradually converges to L max to ensure reliability. The resolution at any given iteration of the optimization algorithm is governed by the discrepancy between the original and current values of the target operating frequencies (cf. "Knowledge-based model fidelity adjustment based on performance specifications"section), and by the algorithm convergence status (cf. "Model fidelity adjustment based on convergence status"section).
Knowledge-based model fidelity adjustment based on performance specifications. Here, we propose to adjust the model fidelity at the initial phase of the optimization run based on the distance D cr =||F cr − F|| between the target operating vector F cr at the current iteration point and the target F. It should be noted that D cr is similar to D c (cf. Table 1), but evaluated using the modified targets instead of the current operating frequencies F c . Further, we denote D cr (0) as the value of D cr after the initial adjustment of the targets, which will be the point of reference for subsequent fidelity modifications. At this point, the fidelity parameter L is set Model fidelity adjustment based on convergence status. The increase of the model fidelity is continued after reducing D cr (i) to zero, i.e., after F cr (i) becomes equal to F. This second stage is governed by the procedure discussed in 79 . It is also assumed that the optimization process is concluded if one of the two conditions is met: (i) ||x (i+1) − x (i) ||< ε x (convergence in argument), or |U(x (i+1) ) − U(x (i) )|< ε U (convergence in the merit function value). Therein, ε x and ε U are the termination thresholds, set to ε x = 10 −3 and ε U = 10 −2 , in the numerical experiments of "Verification case studies" section. Let us also consider the convergence factor 79 10   www.nature.com/scientificreports/ It is employed to decide upon the model fidelity level L (i+1) for the (i + 1)th algorithm iteration. We have where L cr is the model fidelity when the actual frequency F cr (i) first reached the target F. The parameter M determined the convergence level for initiating fidelity adjustment (set M = 10 2 ε x , as recommended in 79 ). Furthermore, the fidelity is obligatorily set to L max near the convergence if the highest L (i) was below L max . In such case, the search region size (cf. "Trust-region-embedded gradient-search" section) is additionally extended by a multiplier M d (we use, M d = 10), and the search continues with L (i+1) = L max 79 . The final acceleration mechanism is to evaluate the Jacobian matrices of the circuit at hand through finite differentiation executed at iteration i at lower level of fidelity L FD (model fidelity for carrying out finite differentiation), rather than L (i) . Here, L FD = max{L min , λL (i) }, with λ = 2/3 (cf. 79 ). It has been observed that this typically results in reducing the overall computational cost (due to good correlation of sensitivities for models of different resolutions), even though the optimization may require a slightly larger number of iterations.

Complete optimization framework
Here, we put together the algorithmic components discussed in "Adaptive design requirements for reliability improvement" and "Variable-fidelity models for optimization cost reduction" sections, and summarize the operation of the entire procedure proposed in this work. The core optimization procedure is a gradient-based routine with numerical derivatives, which will be recalled in "Trust-region-embedded gradient-search" section. "Optimization algorithm" section provides the pseudocode of our algorithm, along with the flow diagram thereof.
Trust-region-embedded gradient-search. The algorithmic components oriented towards improving the reliability ("Adaptive design requirements for reliability improvement" section) and computational efficiency of the search process ("Variable-fidelity models for optimization cost reduction" section) can be incorporated into any iterative optimization procedure. In this work, the core routine is the trust-region (TR) gradient search 80 . The design task is the minimization problem (1). The TR algorithm works iteratively and yields a series of approximations x (i) , i = 0, 1, …, to the optimum design x * as In (10), L (i) is a linear expansion model (3) established at the current design x (i) . Recall that F cr (i) is the current vector of assumed operating frequencies. Throughout the optimization run, the update of the TR search radius d (i) is performed iteratively by taking into account the gain ratio r = [U(S(x (i+1) ),F cr (i) ) − U(S(x (i) ),F cr (i) )]/[U(L S (i) (x (i+1) ),F cr (i) ) − U(L S (i) (x (i) ),F cr (i) )], which quantifies the actual improvement of the objective function (based on EM analysis) versus the estimated improvement (based on the linear model prediction). In case of improvement (r > 0), the design x (i+1) is accepted. Also, if r > 0.75, d (i+1) is increased to 2d (i) ; if r < 0.25, d (i+1) is reduced to d (i) /3. Rejection of the design (r < 0) results in repeating the iteration with a reduced TR size.
Optimization algorithm. The kernel of the knowledge-based optimization procedure introduced in this paper is the TR algorithm briefly discussed in "Trust-region-embedded gradient-search" section. The automated design requirement management strategy of "Adaptive design requirements for reliability improvement" section, and the variable-fidelity model adjustment scheme of "Variable-fidelity models for optimization cost reduction" section, are simultaneously incorporated therein. In particular, both the design goals and the model fidelity are adjusted before each iteration of the TR routine. The goals are modified based on the decision factors of Table 1  and conditions of Table 2, whereas the model fidelity is altered using the coefficient D cr (cf. "Knowledge-based model fidelity adjustment based on performance specifications" section), and the convergence indicator Q (i) (cf. "Model fidelity adjustment based on convergence status" section). Figure 4 shows the pseudocode of the entire procedure, whereas Fig. 5 provides its flow diagram. The designer needs to supply the following information: • Initial design x (0) , • Analytical formula for the objective function U, • Target vector F, • The range of EM model fidelities L min and L max .
Also, the termination condition discussed in "Model fidelity adjustment based on convergence status" section (argument and objective function convergence) needs to be complemented by an additional condition specific to the trust region frameworks, i.e., d (i) < ε x (reduction of the TR size).
The algorithm introduced in "Adaptive design requirements for reliability improvement" through "Complete optimization framework" sections is verified here with the use of three examples of microstrip circuits: two branch-line couplers (a single-and dual-band one), and a dual-band power divider. All circuits are optimized from inferior-quality initial designs, i.e., such whose operating frequencies are away from the design targets. This setup allows us to demonstrate the relevance of the reliability improvements achieved through the adaptive performance requirement approach. At the same time, we investigate computational savings that can be obtained using the variable-fidelity mechanisms incorporated into our procedure. All the simulations were performed on Intel Xeon 2.1 GHz dual-core CPU with 128 GB RAM.

Circuit I: compact branch-line coupler (BLC). The first example is a compact branch-line coupler
shown in Fig. 6a. Figure 6b provides the relevant data, including designable parameters, computational models, initial design, and performance specifications. The circuit is to be optimized to minimize its matching and port isolation, as well as to provide equal power split at the center frequency of 1.0 GHz. The results obtained using the proposed algorithm, standard gradient-based optimization (cf. "Trust-region-embedded gradient-search" section), as well as adaptive design requirements technique 70 , have been gathered in Table 3. The S-parameters of the circuit at the initial design as well as design obtained using the presented approach can be found in Fig. 7 Table 3) that the designs obtained using the algorithm discussed in this work and the method of Koziel et al. 70 are of similar quality. Moreover, the computational speedup achieved through the incorporation of variable-fidelity EM simulations is significant: the total cost of the parameter tuning process corresponds to only 97 high-fidelity circuit analyses (51 percent savings over 70 ). As indicated in Table 3, conventional gradient-based search failed to identify a satisfactory design. The evolution of the design targets and model fidelity has been illustrated in Fig. 8. Note that the major part of the optimization process has been carried out using lower-fidelity models, the high-fidelity simulations are only applied at the latest stages of the algorithm, which translated into the aforementioned speedup.
1. Set the iteration index i = 0; 2. Set model resolution L (i) = L min ; 3. Evaluate circuit characteristics S(x (i) ) and Jacobian J(x (i) ); 4. Find the scalar a to determine current specification vector F cr (i) (a) (cf. Section 2.2); if the conditions D r ≥ D r.min and D c ≤ D c.max , do not hold even for a = 0, go to 7 (premature termination); 5. Perform TR iteration (13) to find the new iteration point x (i+1) according to F cr (i) ; 6. Update the TR radius d (i) ; 7. If D cr (i) = ||F cr (i) -F|| = 0 (current design specifications coincide with original targets) Update model resolution L (i) using (6)   www.nature.com/scientificreports/ Circuit II: dual-band branch-line coupler. As the second verification case, consider a dual-band branchline coupler of Fig. 9a. The important parameters of the circuit have been listed in Fig. 9b. In this case, the design objective is to minimize the matching |S 11 | and isolation |S 41 |, and to achieve equal power split at the operating frequencies of 1.2 GHz and 2.7 GHz. Table 4 gathers the optimization results for the introduced and the benchmark methods. Figure 10 shows the coupler S-parameters at the initial and the final design, x * = [42.0 10.0 0.85 2.56 1.50 1.33 0.60 0.44 2.01] T mm, found by the algorithm of "Verification case studies" section. Similarly as for the first example, the utilization of variable-fidelity simulations leads to considerable computational savings of 61 percent over the adaptive design specification method of Koziel et al. 70 . The cost reduction is achieved without compromising the design quality as indicated in Table 4. In absolute terms, optimization cost corresponds to  Figure 7. Compact branch-line coupler: circuit responses at the initial design (grey lines), and the optimal design rendered by the introduced framework with design specification adaptation and variable-fidelity models (black lines). Vertical line marks target operating frequency.   Figure 10. Dual-band branch-line coupler: circuit responses at the initial design (grey lines), and the optimal design rendered by the introduced framework with design specification adaptation and variable-fidelity models (black lines). Vertical lines mark target operating frequencies. www.nature.com/scientificreports/ 94 EM analyses of the coupler using highest resolution. Figure 11 illustrates the evolution of design goals and model fidelity.
Circuit III: dual-band power divider. The final verification case is a dual-band power divider shown in Fig. 12a. The essential circuit parameters have been provided in Fig. 12b. The aim is to minimize the input and output matching (|S 11 |, |S 22 |, |S 33 |) and port isolation |S 23 | simultaneously at the operating frequencies 2.4 GHz and 3.8 GHz, as well as to obtain equal power division ratio. The latter is implied by the circuit symmetry, therefore, does not have to be explicitly handled in the optimization process. The numerical results are provided in Table 5. The algorithm performance is in accordance with that of the previous examples. On the one hand, we observed considerable computational savings of 54 percent over the single-fidelity procedure of Koziel et al. 70 .
On the other hand, the quality of design produced by the presented method is similar to the benchmark. It should also be noted that the conventional gradient search fails due to severe misalignment between operating frequencies of the coupler at the initial design and the assumed ones. The optimized parameter vector is x * = [26.3 5.09 20.6 5.12 1.0 0.60 4.34] T . The remaining results can be found in Fig. 13 (circuit responses at the initial and optimal designs), and Fig. 14 (evolution of the design specifications and model fidelity). The introduced approach is an accelerated version of the algorithm proposed in Koziel et al. 70 . In contrast to 70 , where only single-fidelity EM model of the component under design is employed, here, we utilize EM  www.nature.com/scientificreports/ models of various fidelities belonging to the continuous range of admissible resolutions. This is a source of the computational benefits of our procedure over that proposed in Koziel et al. 70 . The reliability of our procedure is excellent: it has been capable of yielding the designs fulfilling the required design specifications in all the considered cases, even though the starting points have been to a large extent misaligned with the targets. At the same time, the speedup over the single-fidelity framework 70 is around fifty-five percent on average (from fifty to sixty percent across the benchmark set).  Figure 13. Dual-band power divider: circuit responses at the initial design (grey lines), and the optimal design rendered by the introduced framework with design specification adaptation and variable-fidelity models (black lines). Vertical lines mark target operating frequencies. www.nature.com/scientificreports/

Conclusion
In this work, we proposed a new technique for computationally-efficient and improved-reliability parameter tuning of microwave passive components. The presented approach combines two distinct algorithmic tools, the automated design requirement management scheme, and the knowledge-based adaptively-adjusted EM simulation fidelity mechanism. The former allows for a considerable improvement of the optimization process reliability. In particular, it enables successful local tuning even under challenging conditions (e.g., poor starting point). The latter results in a significant computational speedup with respect to the standard, single-fidelity optimization. Both mechanisms are implemented to work simultaneously. More specifically, the decision-making procedure governing model fidelity setup for a given iteration of the optimization algorithm depends on the current discrepancy between the observed and target operating parameters of the circuit at hand, as well as the convergence status of the search process. The proposed framework is intended to work with full-wave simulation models (e.g., finite-difference time-domain (FDTD), or finite element method (FEM)), but also dedicated solvers that permit a control over discretization density of the structure under simulation. The prerequisite is that the utilized computational models should be evaluated using the same simulation engine, and differ solely by mesh density to ensure satisfactory correlation between the models of different resolutions. Whereas this level of correlation is not possible to achieve with circuit-theory models (or equivalent circuits, or else analytical models). The methodology proposed in this work has been validated using three microstrip components, including two couplers and a power divider. In all cases, it demonstrated superior performance, both in terms of successful allocation of the operating frequencies of the considered circuits despite of poor initial designs, and computational efficiency. The average CPU savings over the recent technique involving adaptively adjusted design specifications are as high as 55 percent. The speedup has been shown not to be detrimental to the design quality. The optimization strategy introduced in this paper has a potential to replace or complement traditional methods, especially in situations where local optimization is likely to fail due to the lack of good starting points or the necessity of re-designing the circuit over broad frequency ranges, whereas the involvement of global search routines is questionable because of the incurred computational expenses.

Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request. Contact person: anna.dabrowska@pg.edu.pl.